----- Original Message -----
>From Vaughan Pratt <pratt at cs.stanford.edu>
Date Wed, 03 Dec 2008 15:15:17 -0800
To Foundations of Mathematics <fom at cs.nyu.edu>
Subject [FOM] Understanding Euclid
asked about how Euclid excluded the geometry of great circles on a sphere.
Euclid's postulate that a line segment can always be extended was
understood to mean "extended to new points," i.e. a segment can always
be extended without returning to itself. That is how he proves
Proposition 16, the first of his propositions that fails in the sphere
geometry: "In any triangle, if one of the sides is produced, then the
exterior angle is greater than either of the interior and opposite angles."
I know that Johann Heinrich Lambert also took the postulate that way,
much later, and I believe pretty much everyone did before the 19th
century.
best, Colin